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Linear Inequalities in Two Variables · Form 4

Linear Inequalities in Two Variables: Practice Questions

Original SPM-style practice questions for Linear Inequalities in Two Variables, each with full worked solutions, Paper 1 multiple-choice and Paper 2 structured.

Written by the spmmath.com.my editorial team.

Original practice questions for Linear Inequalities in Two Variables, in the style of Mathematics Paper 1 (Objective) and Mathematics Paper 2 (Subjective). Each answer is worked so you can check your method, not just the result.

Multiple-choice (Paper 1 style)

Question 1

The inequality y ≤ 2x + 1 is drawn on the Cartesian plane. Which statement correctly describes its boundary line and shaded region?

  1. A. Dashed line, region above the line
  2. B. Solid line, region below the line
  3. C. Dashed line, region below the line
  4. D. Solid line, region above the line

Question 2

Which of the following points satisfies the inequality 3x − 2y > 6?

  1. A. (0, 0)
  2. B. (4, 1)
  3. C. (1, 1)
  4. D. (2, 3)

Question 3

In a hall, x is the number of adults and y is the number of children. The number of adults is at least twice the number of children.

Which inequality represents this?

  1. A. x ≥ 2y
  2. B. y ≥ 2x
  3. C. x ≤ 2y
  4. D. 2x ≥ y

Question 4

A boundary line passes through (0, 4) and (2, 0). The region below the line, excluding the line, satisfies which inequality?

  1. A. 2x + y < 4
  2. B. 2x + y ≤ 4
  3. C. 2x + y > 4
  4. D. x + 2y < 4

Question 5

Which of the following inequalities is drawn with a dashed boundary line?

  1. A. y ≥ x
  2. B. y ≤ 3
  3. C. x + y > 5
  4. D. 2x − y ≤ 0

Question 6

Which point lies in the region defined by the system x ≥ 0, y ≥ 0, x + y ≤ 6 and y ≤ 2x?

  1. A. (1, 4)
  2. B. (3, 2)
  3. C. (5, 3)
  4. D. (2, 5)

Question 7

A feasible region has corner points (0, 0), (4, 0), (2, 3) and (0, 5). What is the maximum value of P = 4x + 3y over this region?

  1. A. 15
  2. B. 16
  3. C. 17
  4. D. 19

Question 8

A feasible region has corner points (1, 1), (5, 1), (5, 4) and (2, 6). What is the minimum value of C = 2x + 5y over this region?

  1. A. 5
  2. B. 7
  3. C. 15
  4. D. 30

Question 9

A workshop makes x chairs and y tables. Each chair takes 2 hours and each table takes 5 hours, and at most 40 working hours are available.

Which inequality models the time constraint?

  1. A. 2x + 5y ≤ 40
  2. B. 2x + 5y ≥ 40
  3. C. 5x + 2y ≤ 40
  4. D. 2x + 5y < 40

Question 10

On the Cartesian plane, which description matches the graph of x ≥ −1?

  1. A. All points on or to the right of the vertical line x = −1
  2. B. All points on or to the left of the vertical line x = −1
  3. C. All points on or above the horizontal line y = −1
  4. D. All points to the right of x = −1, excluding the line

Structured (Paper 2 style)

Question 1 (6 marks)

A tailor makes x school uniforms and y sport uniforms each week. The situation is described by three conditions: she makes at least 20 uniforms in total; the number of sport uniforms is at most 3 times the number of school uniforms; and she makes not more than 15 sport uniforms.

(a) Write three inequalities, other than x ≥ 0 and y ≥ 0, to represent the conditions. (b) Determine whether the point (10, 12) lies in the feasible region by checking each inequality.

  1. (a) 'At least 20 uniforms in total' gives x + y ≥ 20.
  2. (a) 'Sport uniforms at most 3 times school uniforms' gives y ≤ 3x.
  3. (a) 'Not more than 15 sport uniforms' gives y ≤ 15.
  4. (b) Check x + y ≥ 20: 10 + 12 = 22 ≥ 20, true.
  5. (b) Check y ≤ 3x: 12 ≤ 3(10) = 30, true.
  6. (b) Check y ≤ 15: 12 ≤ 15, true. All three hold, so (10, 12) lies in the feasible region.

Question 2 (7 marks)

A bakery makes x kg of butter cake and y kg of chocolate cake in a day. The feasible region for the day's production has corner points A(0, 0), B(20, 0), C(15, 10) and D(0, 18).

The profit is RM 8 for each kg of butter cake and RM 12 for each kg of chocolate cake. (a) Write down the profit function P in terms of x and y.

(b) Calculate the profit at each of the four corner points. (c) State the amount of each cake that gives the maximum profit, and state that maximum profit.

  1. (a) Profit = 8 per kg of butter cake + 12 per kg of chocolate cake, so P = 8x + 12y.
  2. (b) At A(0, 0): P = 8(0) + 12(0) = RM 0.
  3. (b) At B(20, 0): P = 8(20) + 12(0) = RM 160.
  4. (b) At C(15, 10): P = 8(15) + 12(10) = 120 + 120 = RM 240.
  5. (b) At D(0, 18): P = 8(0) + 12(18) = RM 216.
  6. (c) The largest value, RM 240, occurs at C(15, 10).

Question 3 (6 marks)

The boundary of a shaded region is a straight line passing through the points (0, 6) and (4, 0). The shaded region lies below the line and does not include the line itself.

(a) Find the equation of the boundary line. (b) Write the inequality that represents the shaded region.

(c) Determine whether the point (2, 2) lies in the shaded region.

  1. (a) Gradient = (0 − 6) ÷ (4 − 0) = −6/4 = −3/2, and the y-intercept is 6, so y = −3/2 x + 6.
  2. (a) Multiplying by 2: 2y = −3x + 12, which rearranges to 3x + 2y = 12.
  3. (b) The region is below the line and excludes it, so use a strict '<': 3x + 2y < 12.
  4. (c) Substitute (2, 2): 3(2) + 2(2) = 6 + 4 = 10.
  5. (c) Since 10 < 12 is true, the point (2, 2) lies in the shaded region.

Question 4 (7 marks)

A student has RM 24 to spend on x notebooks costing RM 3 each and y files costing RM 4 each. He wants at least 2 notebooks, at least 1 file, and the number of files should not be more than the number of notebooks.

(a) Write four inequalities, other than x ≥ 0 and y ≥ 0, to represent the situation. (b) If he buys exactly 4 notebooks, find the greatest number of files he can buy.

  1. (a) Total cost must be at most RM 24: 3x + 4y ≤ 24.
  2. (a) At least 2 notebooks: x ≥ 2.
  3. (a) At least 1 file: y ≥ 1.
  4. (a) Files not more than notebooks: y ≤ x.
  5. (b) Substitute x = 4 into 3x + 4y ≤ 24: 12 + 4y ≤ 24, so 4y ≤ 12, giving y ≤ 3.
  6. (b) The other conditions y ≥ 1 and y ≤ x = 4 are satisfied by y = 3, so the greatest number of files is 3.

Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)

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Frequently asked questions

What's the difference between the Paper 1 and Paper 2 questions in this linear inequalities set?

Paper 1 questions are multiple-choice, checking whether you can quickly match an inequality to its graphed region. Paper 2 questions ask you to draw the lines, shade the correct region for two or three inequalities together, and state the region clearly, the full working Paper 2 marks reward.

Why should I attempt each question myself before checking the worked solution?

Drawing and shading the correct region for a set of linear inequalities takes practice deciding which side of each line to shade and where regions overlap. Trying it yourself first shows whether you can build that graph from scratch, not just recognise a finished one as correct.

What common mistakes do students make with linear inequalities in two variables?

Students often shade the wrong side of a line, use a solid line where a dashed one is needed (or the reverse), or forget to find the region satisfying every inequality at once. This practice set is designed to catch those slips before the real exam.

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